A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions
Yong Xia Yu-Jun Gong Sheng-Nan Han
Numerical Algebra, Control & Optimization 2015, 5(2): 185-195 doi: 10.3934/naco.2015.5.185
In this paper, by improving the variable-splitting approach, we propose a new semidefinite programming (SDP) relaxation for the nonconvex quadratic optimization problem over the $\ell_1$ unit ball (QPL1). It dominates the state-of-the-art SDP-based bound for (QPL1). As extensions, we apply the new approach to the relaxation problem of the sparse principal component analysis and the nonconvex quadratic optimization problem over the $\ell_p$ ($1< p<2$) unit ball and then show the dominance of the new relaxation.
keywords: $\ell_1$ unit ball Semidefinite programming Sparse principal component analysis. Quadratic optimization

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